An Inertial Forward-Backward Algorithm for Monotone Inclusions

By Dirk A. Lorenz, Thomas Pock. Journal of Mathematical Imaging and Vision, 2014

506 citations recorded by the source corpus.

In this paper, we propose an inertial forward-backward splitting algorithm to compute a zero of the sum of two monotone operators, with one of the two operators being co-coercive. The algorithm is inspired by the accelerated gradient method of Nesterov, but can be applied to a much larger class of problems including convex-concave saddle point problems and general monotone inclusions. We prove convergence of the algorithm in a Hilbert space setting and show that several recently proposed first-order methods can be obtained as special cases of the general algorithm. Numerical results show that the proposed algorithm converges faster than existing methods, while keeping the computational cost of each iteration basically unchanged.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/163ccff8f6da25cec5e649e8aceab33e8ab42720

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Citations in the corpus

The Rate of Convergence of Nesterov's Accelerated Forward-Backward Method is Actually Faster Than 1/k2 (2015, methodology): “This algorithm is closely connected with proximal-based inertial algorithms [1, 14, 23].”
A note on the inertial proximal point method (2015, background): “See also the latest references [11, 21, 14, 12, 13], which analyzed the convergence properties of inertial type algorithms for maximal monotone inclusion problem, variational inequality and structured convex optimization, and demonstrated their performance numerically on some imaging and data analys”
A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces (2019, background): “This improves the weak convergence results of inertial-type algorithms announced by, for instance [33, 40, 52].”
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity (2018, methodology): “This algorithm is within the scope of the proximal-based inertial algorithms [3], [24], [33], and forward-backward methods.”
A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator (2016, background): “The correct general condition has been given in [26, Theorem 5].”
Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists. (2020, background): “Moreover, there are many avenues for extensions, including the use of variable stepsizes or metrics [22,52], the study of convergence rates and development of accelerated variants [104–108], or the design of randomized versions [109–113].”
On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm” (2015, methodology): “The two main differences between our setting and the setting of [11] are: 1.”
On the convergence of the iterates of "FISTA" (2015, methodology): “The sequence (δn)n∈N produced by FISTA, with a good choice of the sequence (tn), has stronger properties than in [9].”
Strong convergence of inertial forward–backward methods for solving monotone inclusions (2021, background): “In 2015, Lorenz and Pock [26] introduced the following inertial forward–backward algorithm (iFBA) for monotone inclusions: { wn = xn + θn(xn − xn−1), xn+1 = (I + λnB)−1(I − λnA)wn.”
An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces (2018, unknown): “Recently, Lorenz and Pock [29] proposed the following inertial forward– backward algorithm for monotone operators: { yn = xn + θn(xn − xn−1) xn+1 = (I + rnB)−1(yn − rnAyn), n ≥ 1, (1.”
A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, methodology): “There have been many works concerning the problem of finding zero points of the sum of two monotone operators (in Hilbert spaces) and accretive operators (in Banach spaces); see [25,26,34,36,48,53,65,66].”